Leggett--Garg Tests of Collective Excitation--Inhibition Dynamics in Neuronal Circuits

Abstract

Recent efforts to identify quantum-like signatures in brain and cognition have explored Bell-type tests, temporal inequalities, and Schrödinger-inspired models. We propose a Leggett--Garg (LG) test of collective neuronal excitation--inhibition (E--I) dynamics. Two reference configurations, $\ket{E}$ and $\ket{I}$, span an effective two-state space governed by $H_S=\heffΩσ_x/2$. Spatial separation of the E and I complexes need not preclude coherent superposition: when they constitute distinct subsystems, the corresponding collective alternatives can form an entangled state. The proposal tests temporal correlations, not Bell nonlocality. Complementary feedforward detectors couple selectively to E or I, with ideal negative results enabling non-invasive inference of the opposite configuration. We give a four-output-cell realization of the required projectors and detector interactions. Ideal sequential measurements yield $K_3(τ)=2\cos(Ωτ)-\cos(2Ωτ)$, reaching $3/2$. We also derive the four-time result $K_4^{\max}=2\sqrt{2}$ and explain its structural CHSH correspondence through Choi--Jamiołkowski duality. Detector efficiency, false positives, imperfect null results, timing uncertainty, and measurement disturbance are incorporated into a realistic extension. We conclude by identifying the experimental challenges of preparing, controlling, and selectively measuring collective E--I states in neuronal circuits.